## H2 z h2 hy h jxy y hz h

If X, Y, and Z are principal body axes, then Eq. (17-41) reduces to

< N cc > 0--p= [Hl„-L,)hIhy+\(Ia-J^)hxh,+Z(Jxx-Jxy)hyhx]

where hx, hy, and hz are the components of the orbit normal unit vector along the principal axes.

From Eq. (17-42), we see that (1) if any principal axis is parallel to orbit normal, the secular gravity-gradient torque is zero and (2) if a principal axis is in the orbit plane, the secular gravity-gradient torque will be along that axis.

The secular gravity-gradient torque for a spin-stabilized satellite can also be calculated from Eq. (17-38). Substituting Eq. (17-35) into Eq. (17-38), the secular torque for a spinning satellite is given by

<N«v^> = —-, / *( 1 + * cos P)(Rs- X Z) dp (17-43)

Writing the unit vector Rs in terms of the true anomaly as R5=fkcosj'+qsini> and assuming that Z, p, and q are constant over one orbit, the average torque is

From Eq. (17-44), we see that (1) the secular torque is perpendicular to Z and therefore does not alter the magnitude of the angular momentum; (2) the gravity-gradient torque causes the spin axis to precess in a cone about the orbit normal with cone angle <J>=arccos(h Z); and (3) the rate of precession of Z is proportional to sin(2\$) and therefore is a maximum at <^=45 or 135 deg.